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Black body
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SQUARE-1<br>Black body
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This puzzle is a cube consisting of three layers. The top and bottom layers are cut like a pie in 8 pieces; 4 edge pieces and 4 corner pieces, 30 and 60 degrees wide respectively. The top and bottom layers can rotate. The middle layer is cut in only two halves along one of the lines of the other layers.
If there are no corner pieces in the way, you can twist half the cube 180 degrees so that pieces from the top and bottom layers mingle.
The puzzle is unique in that the two types of pieces intermingle. The edge and corner pieces can freely move between the two outer layers. Of course, the puzzle will not necessarily be a cube shape if the pieces are mixed. The puzzle has six colours, each face has a single colour similar to the Rubik’s cube.

Square-1 was patented by Karel Hrsel and Vojtech Kopsky on 16 March 1993, US 5,193,809.

The number of positions:
There are three categories of puzzle shapes.
a. Both layers have 4 edges and 4 corners each.
b. One layer has 3 corners, 6 edges, the other 5 corners 2 edges.
c. One layer has 2 corners, 8 edges, the other 6 corners and no edges.

There are 1, 3, 10, 10 and 5 layer shapes with 6, 5, 4, 3 and 2 corners.
This means there are 5*1+10*3+10*10+3*10+1*5 = 170 shapes for the top and
bottom layers. The middle layer has two shapes (half of it is assumed to be fixed). This means that there seem to be 170*2*8!*8! = 552,738,816,000 positions if we disregard rotations of the layers. Some layer shapes however have symmetry, and these have been counted too many times this way.

To take account of the symmetries we can simply count the number of layer shapes differently. Instead of the numbers 1, 3, 10, 10, 5 we use the numbers 2, 36, 105, 112, 54, which are the number of shapes if we consider rotations different (e.g. a square counts as 3 because it has three possible orientations). By the same method as before we then get 19305*2*8!*8! or 62,768,369,664,000 positions. To exclude layer rotations, divide by 122 to get a total of 435,891,456,000 distinct positions.
It is interesting to note that this number is exactly 15!/3. In fact, even for the generalised version with 2n pieces in each layer, a similarly simple formula holds, viz. 4(4n-1)!/3n. I had proved this in a rather complicated unsatisfactory way, but Mike Godfrey found a neat insightful proof click.

He even generalised it to 8(C+E-1)!/(2C+E), where C and E are the total number of
corner and edge pieces, and E>0.
If instead we wish to count only all those positions where there are no corner pieces in the way of twisting the halves, then we can use the same method but counting only all the different ways each shape can be split into two halves, e.g. a square counts as 2 this time. The numbers to plug in are now 1, 12, 46, 62, and 37 which gives a total of 3678*2*8!*8! = 11,958,666,854,400 twistable positions.

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